Elementary mathematical and computational tools for electrical and computer engineers using Matlab - Chapter 7 pot

u=[1 5 3 2]lengthunitu=normunitu %verify length of unit vector 7.2 Dot or Scalar Product If the angle between the vectors and is θ, then the dot product of the twovectors is: 7.8 The dot

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7

Vectors

7.1 Vectors in Two Dimensions (2-D)

A vector in 2-D is defined by its length and the angle it makes with a reference

axis (usually the x-axis) This vector is represented graphically by an arrow.

The tail of the arrow is called the initial point of the vector and the tip of thearrow is the terminal point Two vectors are equal when both their length andangle with a reference axis are equal

The sum of two vectors is a vector constructed graphically as lows At the tip of the first vector, draw a vector equal to the second vector,such that its tail coincides with the tip of the first vector The resultant vectorhas as its tail that of the first vector, and as its tip, the tip of the just-drawnsecond vector (the Parallelogram Rule) (see Figure 7.1)

fol-The negative of a vector is that vector whose tip and tail have beenexchanged from those of the vector This leads to the conclusion that the dif-ference of two vectors is the other diagonal in the parallelogram (Figure 7.2)

If we multiply a vector by a real number k, the result is a vector whose length is k times the length of , and whose direction is that of if k is pos- itive, and opposite if k is negative.

It is most convenient for a vector to be described by its projections on the

x-axis and on the y-axis, respectively; these are denoted by (v1, v2) or (v x , v y)

where ê1 and ê2 are the unit vectors (length is 1) parallel to the x-axis and

y-axis, respectively In terms of this representation, we can write the zero

vec-tor, the sum of two vectors, and the multiplication of a vector by a real ber as follows:

=+ = +

( )( ) ( )( )( )

0 00

r

u2 u12 u

2 2

= +r

2 2 ( ,1 2, , )

7.1.4 MATLAB Representation of the Above Results

MATLAB distinguishes between two kinds of vectors: the column vector andthe row vector As long as the components of the vectors are all real, the dif-ference between the two is in the structure of the array In the column vectorcase, the array representation is vertical and in the row vector case, the arrayrepresentation is horizontal This distinction is made for the purpose ofincluding in a consistent structure the formulation of the dot product and thedefinition of matrix multiplication

You would have observed that:

1 The difference in the representation of the column and row vectors

is in the manner they are separated inside the square brackets

2 The single quotation mark following a vector with real componentschanges that vector from being a column vector to a row vector,and vice versa

3 Multiplying a vector by a scalar simply multiplies each component

of this vector by this scalar

4 You can add two vectors of the same kind and the componentswould be adding by pairs

5 You cannot add two vectors of different kinds; the computer willgive you an error message alerting you that you are adding twoquantities of different dimensions

The MATLAB command for obtaining the norm of a vector is norm Using

this notation, it is a simple matter to define the unit vector in the same tion as a given vector

direc-Example 7.2

Find the length of the vector and the unit vector u = [1 5 3 2] and the unit

vector parallel to it

u=[1 5 3 2]

lengthunitu=norm(unitu) %verify length of unit vector

7.2 Dot (or Scalar) Product

If the angle between the vectors and is θ, then the dot product of the twovectors is:

(7.8)

The dot product can also be expressed as a function of the vectors nents Referring to Figure 7.3, we know from trigonometry the relation relat-ing the length of one side of a triangle with the length of the other two sidesand the cosine of the angle between the other two sides This relation is thegeneralized Pythagorean theorem Referring to Figure 7.3, this gives:

compo-(7.9)but since:

(7.11)and the dot product can be written as:

Case 1: If the vectors are parallel, this means that they make the same angle

with the x-axis The tangent of this angle is equal to the ratio of the vector

x-component to its y-component This means that:

Case 2: If the vectors are orthogonal, this means that the angle between them

is 90°, and their dot product will be zero because the cosine for that angle iszero This implies that:

1 1 2

Solution:Choose two arbitrary points on this line Denote their coordinates

by (x1, y1) and (x2, y2); being on the line, they satisfy the equation of the line:

Substracting the first equation from the second equation, we obtain:

which means that and the unit vector perpendicular

to the line is:

The dot product is written as the product of a row vector by a column vector

of the same length

+ + =+ + =

a x( 2−x1)+b y( 2−y1)=0( , ) (a b ⊥ x2−x y1, 2−y1),

25

15

Observe further, as pointed out previously, the relation between the length

of a vector and its dot product with itself

In-Class Exercises

7.4 for finding the normal to a line in 2-D, to find the unit vector in 3-D that

is perpendicular to the plane:

ax + by + cz + d = 0

(Hint: A vector is perpendicular to a plane if it is perpendicular to two

non-collinear vectors in that plane.)

Pb 7.3 Find, in 2-D, the distance of the point P(x0, y0) from the line ax + by

+ c = 0 (Hint: Remember the geometric definition of the dot product.)

Pb 7.4 Prove the following identities:

r r r r r r r r r r r r r r r

u v⋅ = ⋅v u, u v w⋅ +( )= ⋅ + ⋅u v u w, k u v⋅ ⋅ =( ) (ku v)⋅

7.3 Components, Direction Cosines, and Projections

The components of a vector are the values of each element in the defining

n-tuplet representation For example, consider the vector = [1 5 3 7]

in real 4-D We say that its first, second, third, and fourth components are 1,

5, 3, and 7, respectively (We are maintaining, in this section, the arrow tion for the vectors, irrespective of the dimension of the space.)

nota-The simplest basis of a n-dimensional vector space is the collection of n unit

vectors, each having only one of their components that is non-zero and suchthat the location of this non-zero element is different for each of these basisvectors This basis is not unique

For example, in 4-D space, the canonical four-unit orthonormal basis tors are given, respectively, by:

vec-ê1 = [1 0 0 0] (7.15)

ê2 = [0 1 0 0] (7.16)

ê3 = [0 0 1 0] (7.17)

ê4 = [0 0 0 1] (7.18)and the vector can be written as a linear combination of the basis vectors:

(7.19)

The basis vectors are chosen to be orthonormal, which means that in tion to requiring each one of them to have unit length, they are also orthogonaltwo by two to each other These properties of the basis vectors leads us to the

addi-following important result: the mth component of a vector is obtained by ing the dot product of the vector with the corresponding unit vector, that is,

In 2-D or 3-D, these quantities have the geometrical interpretation of being

the cosine of the angles that the vector makes with the x, y, and z axes.

The projection of a vector over a vector is a vector whose magnitude is

the dot product of the vector with the unit vector in the direction of ,

denoted by ê a , and whose orientation is in the direction of ê a:

Type, execute, and interpret at each step, each of the following commandsusing the above definitions:

7.4 The Dirac Notation and Some General Theorems*

Thus far, we have established some key practical results in real finite sional vector spaces; namely:

dimen-cos(γm)

u u

u

= r = ⋅

rr

r r

r r( ) (= ⋅ ) = ⋅ = ⋅2r

1 A vector can be decomposed into a linear combination of the basisvectors.

2 The dot product of two vectors can be written as the multiplication

of a row vector by a column vector, each of whose elements arethe components of the respective vectors

3 The norm of a vector, a non-negative quantity, is the square root

of the dot product of the vector with itself

4 The unit vector parallel to a specific vector is that vector divided

The Dirac notation represents the row vector by what he called the vector” and the column vector by what he called the “ket-vector,” such thatwhen a dot product is obtained by joining the two vectors, the result will bethe scalar “bra-ket” quantity Specifically:

“bra-(7.23)(7.24)(7.25)

The orthonormality of the basis vectors is written as:

(7.26)where the basis vectors are referred to by their indices, and where δm,nis theKroenecker delta, equal to 1 when its indices are equal, and zero otherwise

The norm of a vector, a non-negative quantity, is given by:

Next, using the Dirac notation, we present the proofs of two key theorems

of vector algebra: the Cauchy-Schwartz inequality and the triangle inequality

The RHS of this quantity is a positive quadratic polynomial in t, and can be

written in the standard form:

at2 + bt + c≥ 0 (7.33)The non-negativity of this quadratic polynomial means that it can have at mostone real root This means that the descriminant must satisfy the inequality:

ifif

22

4 u v 2−4 u u v v ≤0

⇒ u v 2≤ u u v v

which is the desired result Note that the equality holds if and only if the twovectors are linearly dependent (i.e., one vector is equal to a scalar multiplied

by the other vector)

(7.42)

Applying the Cauchy-Schwartz inequality in Eq (7.36) establishes the

desired result The above inequality can be trivially generalized to

n-ele-ments, which leads to the following important result for the equivalent tance for resistors all in series or all in parallel

1 2 3

PROOF The proof is straightforward Using Eq (7.37) and recalling Ohm’s

law for n resistors {R1, R2, …, R n}, the equivalent resistances for this tion, when all resistors are in series or are all in parallel, are given respec-tively by:

combina-(7.44)and

(7.45)

Question:Can you derive a similar theorem for capacitors all in series and all

in parallel? (Remember that the equivalent capacitance law is different forcapacitors than for resistors.)

Homework Problems

geom-etry Parallelogram theorem, which states that: The sum of the squares of the

diag-onals of a parallelogram is equal to twice the sum of the squares of the side; or that:

Pb 7.6 Referring to the inequality of Eq (7.43), which relates the equivalent

resistances of n-resistors in series and in parallel, under what conditions does

the equality hold?

7.5 Cross Product and Scalar Triple Product*

In this section and in Sections 7.6 and 7.7, we restrict our discussions to vectors

in a 3-D space, and use the more familiar conventional vector notation

DEFINITION If two vectors are given by

then their cross product, denoted by is a vector given by:

Pb 7.9 Ask your instructor to show you how the Right Hand rule is used todetermine the direction of a vector equal to the cross product of two othervectors.

As noted in Pb 7.7a, the cross product is a vector that is perpendicular to its

two constituents This determines the resultant vector’s direction To mine its magnitude, consider the Lagrange Identity If the angle between and is θ, then:

deter-(7.50)and

(7.51)

that is, the magnitude of the cross product of two vectors is the area of theparallelogram formed by these vectors

scalar triple product of

PROPERTY

(7.52)This property can be trivially proven by writing out the components expan-sions of the three quantities

7.5.3.1 Geometric Interpretation of the Scalar Triple Product

If the vectors’ original points are brought to the same origin, thesethree vectors define a parallelepiped The absolute value of the scalar tripleproduct can then be interpreted as the volume of this parallelepiped We haveshown earlier that is a vector that is perpendicular to both and ,

and whose magnitude is the area of the base parallelogram From the tion of the scalar product, dotting this vector with will give a scalar that isthe product of the area of the parallelepiped base multiplied by the parallel-epiped height, whose magnitude is exactly the volume of the parallelepiped.The circular permutation property of Eq (7.52) then has a very simple geo-metric interpretation: in computing the volume of a parallelepiped, it doesnot matter which surface we call base.

defini-MATLAB Representation

The cross product of the vectors is found

using the cross(u,v) command.

The triple scalar product of the vectors is found through the

of these functions are defined as 3-D vectors, so that the commands work andthe results make sense

Example 7.8

Given the vectors = (2, 1, 0), = (0, 3, 0), = (1, 2, 3), find the cross uct of the separate pairs of these vectors, and the volume of the parallelepi-ped formed by the three vectors

prod-Solution:Type, execute, and interpret at each step, each of the following mands, using the above definitions:

(1) A great circle is the shortest path between two points on a sphere; (2) theangle between the radial unit vectors passing through each of the cities can

be obtained from their respective latitude and longitude.)

Pb 7.11 Find two unit vectors that are orthogonal to both vectors given by:

Pb 7.12 Find the area of the triangle with vertices at the points:

(1, 1, 1) and is normal to the vector (2, 1, 2)

Pb 7.15 Find the angle of intersection of the planes:

Pb 7.16 Find the distance between the point (3, 1, –2) and the plane z = 2x – 3y.

Pb 7.17 Find the equation of the line that contains the point (3, 2, 1) and is

perpendicular to the plane x + 2y – 2z = 2 Write the parametric equation for

this line

Pb 7.18 Find the point of intersection of the plane 2x – 3y + z = 6 and the line

Pb 7.19 Show that the points (1, 5), (3, 11), and (5, 17) are collinear

(0, 0, 1), (0, 1, 0), and (1, 0, 0)

Homework Problem

ver-tices P0, P1, P2, and P3 are on the unit sphere x2 + y2 + z2 = 1

22

r r r

u v, , andw

u=( , , );2 3 5 v=( , , );2 8 1 w=( ,8 22 12, )

(Hints: (1) Designate the point P0 as north pole and confine P1 to the zeromeridian With this choice, the coordinates of the vertices are given by:

(2) From symmetry, the optimal tetrahedron will have a base (P1, P2, P3) that

is an equilateral triangle in a plane parallel to the equatorial plane The

lati-tude of (P1, P2, P3) is θ, while their longitudes are (0, 2π/3, –2π/3), tively (3) The area of the tetrahedron is the sum of the areas of the fourtriangles (012), (023), (031), (123), where we are indicating each point by itssubscript (4) Express the area as function of θ Find the value of θ that maxi-mizes this quantity.)

respec-7.6 Vector Valued Functions

As you may recall, in Chapter 1 we described curves in 2-D and 3-D by metric equations Essentially, we gave each of the coordinates as a function of

para-a ppara-arpara-ameter In effect, we generpara-ated para-a vector vpara-alued function becpara-ause theposition of the point describing the curve can be written as:

(7.53)

If the parameter t was chosen to be time, then the tip of the vector would

be the position of a point on that curve as a function of time In mechanics,finding is ultimately the goal of any problem in the dynamics of a pointparticle In many problems of electrical engineering design of tubes and othermicrowave engineering devices, we need to determine the position of elec-trons whose motion we control by a variety of electrical and magnetic fieldsgeometries The following are the kinematics variables of the problem Thedynamics form the subject of mechanics courses

To help visualize the shape of a curve generated by the tip of the positionvector , we introduce the tangent vector and the normal vector to thecurve and the curvature of the curve

The velocity vector field associated with the above position vector isdefined through:

P P P P

R t( )

r

R t( )